Infinite Horizon Noncooperative Differential Games with Non-Smooth Costs

both hi being integrable functions, whose smoothness will be addressed later. Very few results are known on the subject, except in two particular cases: two players zero-sum games and LQ games (where LQ stands for linear-quadratic). Indeed, a key step in this kind of problems is the study of the value function u. In the region where u is smooth, its components satisfy a system of HamiltonJacobi equations (see [7]), and this system is usually difficult to solve. In the case of two players zero-sum games, since what one player gains is exactly what the other player loses, the two components of u are one the opposite of the other. Hence, the Hamilton-Jacobi system (HJ in the following) reduces to a single equation and one can apply the standard theory of viscosity solutions (see [1] for more details) to obtain existence and uniqueness results. In the case of LQ games, the HJ system can be connected to a Riccati system of ODE for matrices. This system is in general much easier than the original one, and standard ODE techniques can be applied (see [6] for a detailed treatment). On the other hand, in the present case both approaches fail and therefore one has no established techniques to rely on. However, few results still can be proved. In the finite horizon setting, the analysis presented in [3, 4] showed that, for a non-cooperative n-players differential game with general terminal payoffs, the well-posedness is strongly related with the HJ system being hyperbolic. Namely, for x ∈ R, thanks to recent advances in the theory of hyperbolic systems of PDE, games with strictly hyperbolic HJ systems are well-posed. On the other hand,